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Homework Help: Help with Subgroups

  1. Sep 21, 2010 #1
    1. The problem statement, all variables and given/known data

    if H and K are arbitrary subgroup of G, prove that HK=KH iff HK is a subgroup of G

    2. Relevant equations

    n/a

    3. The attempt at a solution

    no problem to prove => direction

    for <= i can prove KH is a subset of HK

    only i got troubled to show HK ia subset of KH

    x in HK

    x=hk for some h in H ,k in K

    i manipulate it many ways and always got the form x=khk for some h in H ,k in K

    HELP, and sorry no latex ,i'm very buzy now ;P
     
  2. jcsd
  3. Sep 22, 2010 #2
    Re: subgroups

    Take b in HK so that b-1 = hk is in HK. How do you get b back?
     
  4. Sep 22, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    Re: subgroups

    so [itex] k^{-1}h^{-1} \in KH[/tex]

    and [itex] hk \in HK [/itex], what's its inverse?
     
  5. Sep 22, 2010 #4
    Re: subgroups

    aahhhh i see,

    b in HK

    b=hk for some h in H k in K

    b^{-1} also is in HK

    imply
    [itex]
    b^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH
    [/itex]

    right ??
     
  6. Sep 22, 2010 #5
    Re: subgroups

    wait wrong,

    for any x in HK, x^-1 in HK so x^-1=hk for some h and k

    then [itex]

    x=(x^{-1})^{-1}=(hk)^{-1}=k^{-1}h^{-1} \in KH

    [/itex]

    now this is correct right?
     
  7. Sep 22, 2010 #6

    lanedance

    User Avatar
    Homework Helper

    Re: subgroups

    yeah that looks good
     
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